So I made this guide for the NS Photographers Cult, but since some of you aren't in it, I thought it might be worth throwing out here to help more of you! You should still join the cult though :)
https://www.newschoolers.com/ns/cult/forum/cat_id/6302/**DISCLAIMER**
This post may not be 100% accurate, truthful, or use "proper" terminology. I'm going to try and phrase everything as simple as possible so everyone can benefit from the thread. It may contain personal bias, but hey, we're all human :) Oh, and I'm mostly going to talk about digital SLR's with Focal Plane Shutters since they are the most used on this forum and are easiest to explain.

Section I - The Basics

The Strobe, or Flash -The flash or strobe is your basic tool for controlled lighting of photographs, and its task is simple: to emit a pulse of bright light to help expose your subject in a photo. How does this magically happen when you press the shutter button? Well, the camera sends an electronic signal to either the hotshoe or PC sync port telling the connected flash to fire. This timing usually coincides with the sensors or films full exposure from the shutter (i.e. when the first shutter curtain is open all the way and the second curtain is not yet closed). There are many different kinds, sizes, and powers of strobes, which we will talk about in detail later.

Section II - The Camera

Familiarize yourself with your camera and its ability to use flashes. The most standard flash you'll see is the on camera flash. It's the one that usually pops up from above the lens mount on most consumer and some prosumer models. it looks something like this:
(fig 2.1 on camera flash)
next we have the hotshoe. It's used to attach either an on camera speedlight, a radio trigger (discussed in a bit), a hot shoe extension cord, or a hotshoe to PC sync adapter (these can be used incase your camera does not have a PC sync port and you'd like to use one. they can be found cheap on ebay). Your camera hotshoe looks like this:
(fig 2.2 hotshoe connector)
(fig 2.3 pc sync port)
The shutter of your camera is very important to understand when doing flash photography. The majority of you have Digital Single Lens Reflex cameras (DSLR) with a focal-plane shutter. These shutters work by blocking light off from the sensor, and open up to expose the sensor to the light for a certain amount of time that you determine before quickly closing back up. All shutters have a maximum sync speed (xsync), which is the fastest shutter speed that allows the full sensor to be uncovered at one time, without the shutters blocking of either the top or bottom of the sensor. Watch the video below to get an idea of how a fast (1/1000th of a second) shutter speed works vs. a 1/200th.

(vid 2.1 slowmotion camera shutter)

Section III - Flash Units

Hotshoe Flashes - Hotshoe flashes, or speedlights, are the most common and probably the most used flashes. They fit into your cameras hotshoe or can be mounted pretty much anywhere your creativity limits them to. They are usually weaker than monolights, but are way more portable. Coming in at all different shapes and sizes, there is usually a pretty simple and standard design for the more professional flashes, looking something like this:
(fig 3.1 hotshoe flash)
Monolights - Monolights, or studio strobes, are larger flash heads that provide more powerful light output and greater use of modifiers at the cost of portability. Also coming in many different shapes and sizes, they usually have a head that is mounted on a lightstand, and are AC powered or have a power pack attached to them via cable.
(fig 3.2 studio strobe)
Each flash system has its pros and cons. Depending on what you are shooting, how light it is out, or wether you are in a controlled environment, each system can be used in their own ways. I'll focus on sports shooting (seeing that most of us are skiers...).

Section IV - Triggering your strobes

There are several ways to trigger your flash, depending on how much flexibility or control of the flash you want. Here are some of the most common ways of triggering:
Direct connection of your flash to you camera involves either having your camera set in the hotshoe, or having some wires lead to the flash from the PC port, hotshoe, or hotshoe adapter to your strobe. Probably the most reliable method of triggering, it is also the most cumbersome and least flexible.
Radio slaves, such as pocketwizards, cactus, cybersyncs, and ebay triggers, all use radio frequencies in a transmitter connected to your camera to wirelessly trigger a receiver unit attached to your strobe. These are probably the second most reliable, and most flexible way of triggering.
(fig 4.1 pocketwizard plus sytem)
Optical slaves "see" light from another flash unit and tell the connected strobe to fire. They rely on direct line of sight to the master flash unit, and can be effected by sunlight and other peoples flashes. Best used in a controlled studio.
(fig 4.2 optical peanut slave)

Section V - Using strobes pt I (single strobe usage)

Strobes are incredibly handy for creating a controllable light source. They can be used as a single light source, used to balance or overpower ambient light, or used in groups to create very professional looking images. In this section I'll be covering single strobe usage.
Single strobes are typically used by combining ambient (the light naturally available in the scene) lighting and the strobe's light to help enhance the lighting on the subject. They can also be used as backlights, fill, or used creatively to highlight a certain part of an image. It's hard to really say how to use them, as every photo setup is different, and up to the photographer to determine mood/style/emotion they want to portray in the final image. Probably best to use example images to show, rather then text. Much easier ;) EDIT: for a great collection of single strobe shots, I suggest checking out this thread:
http://photography-on-the.net/forum/showthread.php?t=969584
(link 5.1 photography-on-the.net post your creative single strobe shots thread)
The picture below (left) illustrates a primary flaw to pop-up and on camera flashes pointed directly at the subject: they "flatten" out the image and fail to highlight depth of the subject. As the flash is moved off camera left, you can see the dramatic difference and change in the image; the model now appears separate from the background, and the lighting is more "normal" looking. (Yes, the shadows are harsh, but refer to section VII [Multiple Strobe Usage] or section IX [Light Modifiers] for ways to fix that!)
(fig 5.1 on camera vs off camera flash)
(fig 5.2 "Giulia and Some Thoughts")
I chose the above image for two reasons. The first is the great use of the strobe to highlight our model and create a focal point in the image. The second is more thanks to some modification of the light (section IX) that creates a nice fade of the light against the background columns, and the more abrupt drop off in the foreground to create little distraction.
(fig 5.3 model and camaro)
This image (above) is a textbook example of using a strobe to overpower natural lighting. Very simple: Sun (high camera right and opposite of models facing direction) is used as both a back light (on the model) and primary light (background, most of the car), while the flash is used to eliminate the shadow caused by Mr. Sun on the model and car.
Last we have an absolutely wonderful use of a simple hotshoe speedlite off camera. To the untrained eye, this photo would appear to be captured with natural lighting. However the ever-so-slight kiss of the strobe eliminates any harsh shadows and underexposure the bright background makes, while not blurting out the "I used a flash in this image" statement.
(fig 5.4 wedding couple w/ strobe)

Section VI - Exposing your image (mainly for sports/action stopping)

Making sure to properly expose your image with a flash can be difficult. However, once you get the hang of it, it can be simple. I personally am a trial and error kind of guy, and will take test samples until I dial in the proper exposure. But using my knowledge of how cameras work, I can easily make sure to get my desired flash effects. Here are a few handy tips for exposing flash photography:
Aperture controls the exposure of BOTH the subject and ambient light. Just like a regular photo, an aperture of f8 will be darker than an aperture of f4. After placing your flash and choosing its power level, the BEST (and usually only) way of dialing in your subject's exposure is by adjusting aperture.
Shutter Speed, contrary to popular belief, will ONLY change the exposure of the ambient lighting. Regardless of how fast or slow it is, the duration of the flash and its power level will still illuminate the subject the same, and the sensor will still absorb the same amount of light from it. However, the longer the shutter is open the more light from the ambient can come in (this can effect the rider's before/after flash exposure). therefore, the BEST way to control ambient lighting is through shutter speed.
Location, location, location. Location of your flash plays a huge part in exposing your subject(rider). The closer the flash is to the rider, the more it will effect them. But REMEMBER THIS: the power of your flash decreases exponentially as you move it farther away. Many people believe that doubling the distance from the subject will decrease the power by 1/2. However, it actually decreases by 1/4th (this is known as the inverse square law). You can read more about it in this guide:
http://photography-on-the.net/forum/showthread.php?t=171657
(link 6.1 flash photography 101)
Your flashes power level should also be taken into account. Depending on your flash manufacturer and type of flash (monolight vs speedlight), you'll either get your fastest duration at maximum power or minimum power. RESEARCH YOUR FLASH MODEL TO FIND OUT!! That way, you know wether to place your flash a bit farther from your subject if this occurs at full power, or close if it occurs at minimum. I'll talk about durations more in Section X.

If you are looking to freeze action, you have to remember that you are most likely dealing with some form of ambient light (the sun, indoor lights, etc). Remember that this DOES still effect your subject! You'll have to account for it when bringing a flash into the picture, and remembering how a flash freezes action, you'll most likely need to underexpose the ambient. Why? Well, if you have a properly exposed skier moving from left to right, with a shutter speed of 1/200th and no flash, you'll get a pretty blurry image as the action is highlighted for the entire exposure. When a flash is brought in, even though it would burn in a specific part of the skiers action into the film/sensor, a trail and motion blur would still form behind or in front of the skier. This is because the ambient is still lighting the skier like a regular photo! Now if we compensate for the ambient lighting by underexposing it, the rider will only be lit at proper exposure by the flash, and his motion blur will be significantly reduced. To further help avoid motion blur, track your subject as to "stack" the riders image into one specific part of the sensor.

Section VII - Flash durations, explained

Ahh, as a skier and sports shooter you probably are wondering how the hell you are going to freeze the action with a limited shutter speed. Well, I'm here to tell you the flash will do it for you (nifty, huh?). Flashes emit a pulse of light in a very short amount of time (Called the duration). Depending on the flash brand/make/model and the power setting it is on, this duration can either be very, very fast (1/6000th+) or slower (1/250th-). Recalling how an image is exposed (the sensor absorbs light for your desired period [shutter speed]), lighting your subject with a bright pulse of light will "burn" that image into the exposure. The faster the duration of the flash (assuming it is close and bright enough to light your subject properly), the better the action will be captured, just like using a higher shutter speed.

(vid 7.1 basic flash duration video)

Section VIII - Using strobes pt II (multiple strobe usage)

Some of you may shoot more than just sports, or would like to bring multiple strobes into the field, which is always great. Using more than one strobe helps add a "3D" affect to your image and makes separate subject from background. The most common use of multiple flashes is in portraits, which I will discuss first.
For portraits, using several strobes helps light your subject, remove shadows, and create depth. Here is a typical three light setup:

Your Key Light is used as your main light source. Most commonly, it is placed just off-set to your cameras left or right, however you can place it wherever you like. Its purpose is to light most of your subject and set the mood of the picture.
The Fill Light is used to remove shadows that are created by the key light. Usually set a little less powerful and opposite the key, as to keep a little bit of shading (for depth and highlighting of features).
Finally, your Back Light (or Rim Light) is used to separate the subject from the background and highlight the outlines of whatever you are shooting.

If you want to fool around with different light setups, but dont have the equiptment to, try an ONLINE SIMULATOR, LIKE THIS ONE! Seriously, I just found this finishing up this guide and it is AWESOME.
---- http://www.zvork.fr/vls/ ----
(link 8.1 headshot simulator)
This simulator is good too: http://www.mediacollege.com/lighting/three-point/simulator.html
(link 8.2 lighting simulator)
We can also bring this information and use it for sports as well. Shadows are usually harsh when using a single strobe, and bringing another one (or more) to fill those shadows will create a much more natural and balanced image. Here are a couple of examples of using multiple strobes:
(fig 8.1 "bebo (1 of 1) album cover")
(fig 8.2 "Fabian Surber - Bs Flip")
(fig 8.3 "Philadelphia, PA May 2010")

Section IX - Light modifiers

There are so many modifiers out there that this section will be just a brief touch on some of them. But in the end, all of them boil down to one simple principal: they modify the light your flash puts out. Wether they soften it, diffuse it, narrow it, control it, or color it is up to your choice. Here are a few basic and more common ones.
Snoot - A snoot is a narrow tube that restricts light from your flash to a certain area in a beam. usually not reflective on the inside, all it does is prevents the light from "bleeding" out and creates a nice (but rather harsh) spotlight effect.
Gel - Gels are sheets of clear, usually colored or tinted, material that you place in front of your flash to color that light. Good for accents or creating a mood!
Umbrella - Very common among the strobist community for being cheap and easy to transport. The two most popular uses of umbrellas are shoot-through (a white umbrella that diffuses the flash into a more gentle and soft light) and reflective (usually black on the outside and reflective on the inside, they face open to the subject and allow the flash beam to spread from more narrow to a wider and even beam)
(fig 9.1 umbrellas)
Reflector - There are two I'd like to talk about. The first is one that attaches to a flash head, usually with the flash tube/bulb somewhere inside the cone shape, that control how wide or narrow the beam is coming directly from the unit. There are reflectors that are narrow and constrict the beam to be narrow and bright, or some that are wide (which do the opposite!)
The other reflectors (sometimes called bounce cards) are used to re-direct light. These are great when you are limited to a few lights and would like to add another source. Simply bounce "wasted" or "excess" light back at your subject (for example, shooting against the sun with one strobe, use the light from the sun bounced against a reflector to use as a fill against the shadows created by your key strobe). 5 in 1 reflectors are one of the handiest tools to have and because you can get them cheap they are a no-brainer to have in a kit (see section X for more on them).
(fig 9.2 5in1 reflector)
Softbox - Softboxes are diffusers and reflectors built into one modifier. They come in pretty much any size and shape you can think of, but all do something very similar: reflect light inside the box, direct it towards the subject, then diffuse it through a semi-translucent skin attached to the front. The fall-off on softboxes is a little more direct than a shoot through umbrella (as the umbrella is curved and diffuses in an arc vs beam manner).
(fig 9.3 softbox)

Section X - Buyers guide!

Researching which flash to buy can be very, very confusing. here are a couple of tips and hints to help you make the right purchase.
For sports shooting, flash duration is going to make or break your purchase decision. Manufacturers list two different duration numbers, called t.1 and t.5, which are measurements of the duration the flash is lit up until 10% of its peak output, and 50%, respectively. Here is a chart that shows the t.5 and t.1 durations. (The black line shown is the flashes output in regards to time. It gets up to its peak very quickly, then falls off exponentially)
(fig 10.1 flash duration chart)
When buying a flash for action stopping, you want to refer to the t.1 time! The t.5 was created for the sole purpose of tricking people into thinking the flash's duration was quick. While it does show that the flash will power up and fall off to 50% in a quick amount of time, you still will get exposure for the remaining 49% down to 10% AND YOU NEED TO ACCOUNT FOR THIS. Make sure you can get something with a t.1 of 1/1000 or faster at or close to full power. Skiing will require the unit to be rather far from your subject, and getting enough power (refer to inverse square law) while keeping a fast duration will be key.
If you are looking for maximum portability, a speedlight is going to be the flash for you. I recommend going third party on one, as Canon and Nikon's both are expensive for what they offer. High end Speedlights are known for a feature called TTL (Through The Lens) metering, where the camera tells a flash to stop firing once a proper exposure has been reached. This is a nice thing to have on the fly, but I advise to learn to strobe full manual (as most studio strobes will be manual anyways). Make sure the model you have allows for manual adjustment of power settings! I also recommend getting as powerful a flash as possible, as most speedlights are not strong to begin with and most have fast durations. Ebay is a great starting point. Look for Yongnuo, Nissin, Vivitar, Sunpak, older Nikon (2x instead of X00 series I believe)
***NOTE ON OLDER MODEL STROBES***
Buying old flashes can be a great way to save money and get a great product. However, some older flashes have insanely high trigger voltages which can fry your camera if placed in your hotshoe. READ THIS GUIDE AND MAKE SURE THE FLASH IN QUESTION IS SAFE FOR USE. It will save your camera (and yourself) from a very bad time. http://www.botzilla.com/photo/strobeVolts.html
(link 10.1 trigger voltage list)
As for monolights + skiing, you are going to need a pack and head style kit (or some sort of AC adapter like Paul C. Buff offers). There are several options, most being $2000+. Most talked about/positively reviewed are Elinchrom, Alien Bees (Paul C. Buff), Lumadyne, Broncolor, Profoto, and Dynalite. The Alien Bees/Paul C. Buff systems are the cheapest of the lot (by a large margin) but are fantastic to use. As of right now I own a b400 and a Vagabond II power pack, and am looking to upgrade to an Einstein unit. I'd recommend an einstein or b800 for a head, as both have decent power output while keeping fast durations. Elinchroms are used by many professional ski photographers and have a great reputation for being both powerful and durable in the field. Profoto has a great 600ws setup for just a touch over 2300, with durations ranging from 1/1000th to 1/6800. If you have the money, research reviews and look over those durations to decide which of the higher end setups is right for you.
Triggers - going wireless is a great thing to be able to do. Rigging your flash in a hard to get location and you'll be glad you had some. Choosing your brand/model will depend mostly on your budget. You'll need a transmitter unit and a receiver for every flash you intend to fire (unless you go something that has transceiver units, which function as both in one). "ebay" triggers are usually pretty cheap for a setup, and work decently. They do not have the range, durability, or reliability of an expensive setup, but are good to have if you are putting a flash in a risky location or are really budget stricken. Cactus, Pocketwizard, Cybersyncs, Skyports are the bigger-name and more expensive units, but are all known to be consistant and dependable. Pocketwizards (pictured below) are very notorious for being the top of the line, professionally used systems that have a fantastic range of 400m+ in some cases.
(fig 10.2 pocketwizard plus II system)
There are a couple of chinese replica companies, such as Pixel, Phottix, and others that also make cheaper variations/knockoffs of PW and others, and have done a good job making a decent product accessible to the common man.
A note with triggers though is that some support the TTL metering I mentioned above and will usually be priced much much more expensive. I suggest not buying these (unless trying to get hypersync) as you will most likely be using a manual flash anyways. Not worth the extra $$$!
AccessoriesNot going to go too in depth here, but some of the "must buys" and essentials to a good lighting kit need to be covered. Lightstands are essential, and most of them out there are pretty good. I suggest: reading reviews on b&h, and plan to spend $30-60 on a nice 9'-13' stand (more if you'd like air cushioning). Spending under that is silly, you'll get a flimsy product that will break (or worse, break your lights!). If you are planning on shooting speedlites, you'll need a hot shoe umbrella bracket. (googling it will give you a million different ones to buy. Again, spend a little more on something well reviewed).
Also suggested would be:
-A 5 in 1 reflector: $25/35 max on amazon or ebay, any size you feel you need. SO HANDY!
-A shoot through and bounce umbrella
-gaffers tape: strong as duct tape, but doesn't leave residue.
-an set of ebay gels for your speedlight: lets you get very creative with an accent or primary flash. maybe $8-10 from china via ebay. great thing to have!
-ball bungies: strap your speedlight to anything you can wrap one around in a pinch.
-extra batteries!

Section XI - Conclusion
aright, wrapping this up (finally). There are probably lots of typos, poor grammar, and contradictions in the post, but it's taken me a couple of months to put together and I'm sure I've missed something in proofreading it! Hopefully you find it helpful, and post your comments/questions/advice/hints for others below.

Strings are really helpful when shooting skateboarding, travelling with light stands is a bitch and they often end up in the photograph (This is a no-no) when your shooting fisheye. I also really like using a gorilla pod as a light stand, its great for travelling light and getting flashes in places a light stand wouldn't work.

As for triggers and flashes, yongnuo's are amazing for strobist stuff. I have the rf-602 trigger/receivers and two yn-560's. They are a bit slow for action, but I'm 95% sure they have a faster flash duration than the acclaimed vivitar 285hv. They have a great, crisp light, with a white umbrella they are perfect for low key portraiture (my favourite). Cactus triggers are also really good if your on a budget.

I'm also experimenting with fully flash lit sequences right now, I'll make another post when I get some results.